# BOUNDARY VALUE METHODS FOR NUMERICAL SOLUTION OF FOURTH ORDER PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONS

Thesis

Many real life situations can be modelled mathematically as Ordinary Di er-ential Equations (ODEs) or Partial Di erential Equations (PDEs). Problems arising from these are either of Initial Value Problem (IVP) or Boundary Value Problem (BVP) types in the case of ODEs and Initial Boundary Value Problems (IBVPs) in the case of PDEs. As PDEs are more di cult to solve, techniques have been developed to transform them into ODEs of BVP form. For higher order di erential equations, their solutions are mainly obtained numerically by reducing them to equivalent rst order systems. However, this is not considered e cient if direct method could be found for their so-lution. The main focus of the present study is therefore the development, analysis and application of a class of Boundary Value Methods (BVMs) for direct solution of Fourth Order ODEs of both Initial Value and Boundary Value types. A BVM is a Linear Multistep Method (LMM) coupled with boundary conditions. Also BVMs have the advantage of being self - starting viz-a-viz many other numerical methods. Therefore, the objectives of this study were to: (i) develop polynomial- tted BVMs for solving fourth order ODEs; (ii) develop trigonometrically- tted block BVMs for solving oscilla-tory fourth order ODEs; (iii) analyse the basic properties of the methods developed and these include zero- stability, consistency and convergence of the methods; (iv) implement the methods on speci c fourth order ODEs; and compare the performance of the proposed methods with those of existing ones. The problems considered in this work were the IVP : v yiv= f(x; y; y0; y00; y000) a x b y(a) = y0; y0(a) = y00; y00(a) = y000; y000(a) = y0000 and the BVP : yiv= f(x; y; y0; y00; y000) a x b y(a) = A1; y0(a) = A2; y00(b) = B1; y000(b) = B2 Trial solution of the type: p+q 1 Xr U(x) = arxr' y(x) =0 for the polynomial - tted BVMs and p+q 3 Xr arxr+ ak+3cos wx + ak+4sin wx ' y(x) U(x) = =0 for trigonometric - tted BVMs, where ar are uniquely determined coe cients,! is the frequency, and p and q are the number of interpolation and collocation points, respectively. Interpolation and collocation were achieved through the sets of equations: u(xn) = yn; u(xn+1) = yn+1 u(xn+k) = yn+k uiv(xn) = fn; uiv(xn+1) = fn+1; uiv(xn+k) = fn+k: This system of equations was solved by Mathematica 8.0 for ar; r = 0(1)(k+ The substitution of the resulting a0rs into the trial solution yielded the desired BVMs. The ndings of the study were the: vi derivation of BVMs with polynomial and basis; derivation of BVMs with trigonometric basis ; derived methods are consistent, zero stable and convergent; proposed BVMs that can handle both initial and boundary value prob-lems; and methods compare favourably well in terms of accuracy with existing methods . The study concluded that a class of BVMs with polynomial and trigonomet-ric bases derived was developed and successfully implemented on sti and non- sti problems in fourth order ODEs. It is therefore recommended for application in determining the solution of real life problems leading to IVPs and BVPs in Fourth Order ODEs.